We investigate necessary and/or sufficient conditions for the pointwise and uniform convergence of the weighted Hankel transformsWe subdivide these transforms into two classes in such a way that the uniform convergence criteria is remarkably different on each class. In more detail, we have the transforms satisfying µ+ν = 0 (such as the classical Hankel transform), that generalize the cosine transform, and those satisfying 0 < µ + ν ≤ α + 3/2, generalizing the sine transform. *

We obtain necessary and sufficient conditions for the uniform convergence of sinewhere satisfies general monotonicity conditions. In contrast with the previous results on this topic, here we do not assume ≥ 0. K E Y W O R D SFourier transform, general monotone functions, sine transform, uniform convergence M S C ( 2 0 1 0 ) Primary: 42A38; Secondary: 26A45, 26A48, 40A10 www.mn-journal.org

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel-Olivier test for realvalued functions. Analogous results for cosine series are derived as well. We also show that our statements do not hold without the general monotonicity assumption in the case of cosine integrals and series.